A Two-Step Approach for Optimal Control of Kinetic Batch

7/27/2019 A Two-Step Approach for Optimal Control of Kinetic Batch

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Paweł DRAG1, Krystyn STYCZEN 1

Wrocław University of Technology, Insitute of Computer Engineering, Control and Robotics (1)

A Two-Step Approach for Optimal Control of Kinetic BatchReactor with electroneutrality condition

Abstract. In the article a Multi-Step Solution Technique were used. This technique enables to translate a complex optimal control problem to easier nonlinear programming problems. Efficiency of the designed algorithm was verified on a real-live optimal control problem of a stiff and highly nonlinear differential-algebraic system with constraints. One of the algebraic state equations takes the form of an electroneutrality condition, which ensures failure-free run of the production process. The simulations were executed in MATLAB, which is a commonly known programming environment.

Streszczenie. W artykule wykorzystano technike wieloetapowego poszukiwania rozwiazania. W ten sposób zło˙ zony problem sterowania optymalnego mo zna traktowa c jako grupe prostszych problemów programowania nieliniowego. Działanie wykorzystanego w pracy algorytmu najlepiej wida c na problemie doboru optymalnego sterownia dla zło zonego układu opisanego za pomoca nieliniowych równa n ró zniczkowo-algebraicznych z ograniczeniami. Jedno z algebraicznych równa n stanu przybiera posta c warunku elektronetralno sci zapewniajacego bezawaryjny przebieg procesu.Symulacje zostały przeprowadzone w ´ srodowisku MATLAB. (Dwustopniowe podej scie do sterowania optymalnego reaktorem wsadowym z warunkiem elektroobojetno sci)

Keywords: optimal control, sequential quadratic programming, DAE systems, multiple shooting methodSłowa kluczowe: sterowanie optymalne, sekwencyjne programowanie kwadratowe, układy równan rózniczkowo-algebraicznych, metodawielopunktowych strzałów

IntroductionThe optimal control problems are understood as a

searching for optimal control trajectory, which minimizes a

cost function for specified class of dynamical system. Ex-

pression ¨dynamic systems¨ one can found as structures,

which can be described as a system of ordinary differential

equations (ODEs). Another kind of dynamic systems are hy-

brid systems [1] [2]. There are ODEs, which contain switch-

ing instants. Specified type of optimal control problems was

introduced together with using of periodic control [3]. When

the article ¨Differential/algebraic equation’s are not ODEs

[4] was published, many scientists around the world concen-

trated their attention in the new class of systems.

Differential-algebraic systems consist of two parts.There are differential equations in the first part, and algebraic

equations in the second part. Typical problems, which can be

expressed as DAEs, are Constrained Variational Problems,

Network Modeling and Discretization of PDEs. The authors

of one of most important books about Differential-Algebraic

Equations ended their publication with the words ¨Applica-

tions such as mechanical systems, electrical networks, tra-

jectory prescribed path control and method of lines solution

of partial differential equations have had an enormous im-

pact on the development of both the numerical analysis and

mathematical software for DAE systems, and will undoubt-

edly continue to influence the future developments in this

area¨. [5]

One of the advanced problem in chemical engineeringis optimal control of Kinetic Batch Reactor. Kinetic Batch

Reactor is described as an nonlinear and stiff DAE system

[6, 7].This problem illustrates the interference of dynamic

equilibrium conditions with various physical meaning. For ex-

ample electrical charges and chemical phases. Difficulties

associated with the searching for a solution led to the devel-

opment of a new algorithm. To solve this problem a Two-Step

Solution Technique was used. This method enables to solve

this problems without using highly specialized codes, but only

using popular, coupled to the MATLAB R2010b environment,

solvers.

Optimal control of DAE systemsIn the paper the following DAE optimization problem was

considered

(1) ϕ( p) =

N T

l=1

Φl(yl(tl), zl(tl), pl) → min,

subject to

(2)dyl

dt= f l(yl(t), zl(t), pl)

with initial conditions

(3) yl(tl−1) = yl0

and algebraic equations

(4) gl(yl(t), zl(t), pl) = 0, t ∈ (tl−1, tl], l = 1,...,N T

with bounds

(5) plL ≤ pl ≤ plU ,

(6) ylL ≤ yl ≤ ylU ,

(7) zlL ≤ zl ≤ zlU

and constraints

(8)

h( p1,...,pN T , y10, y1(t1), y20, y2(t2),...,yN T

0 , yN T (tN T )) = 0.

There are differential variables y(t) and algebraic variables

z(t) in the DAE system in semiexplicit form. The assump-

tion of invertibility of g(−, z(t),−) permits an implicit elimina-

tion of the algebraic variables z(t) = z[y(t), p]. While there

are N T periods in DAE equations, time-dependent bounds

or other path constraints on the state variables are no longer

considered [8]. The control profiles are represented as pa-

rameterized functions with coefficients that determine the op-

timal profiles [9, 10]. The decision variables in DAE equations

appear only in the time independent vector pl. Algebraic con-

straints and terms in the objective function are applied only

at the end of each period.

The problem 1-8 can be represented as the following

nonlinear program

(9) φ( p) → min

176 PRZEGL AD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 88 NR 6 / 2012



subject to

(10) cE ( p) = 0,

(11) cI ( p) ≤ 0.

In the literature one can find a lot of methods solving above

problem (see [6, 8, 11, 12, 13, 14]).

Sequential dynamic optimization strategyA sketch of the sequential dynamic optimization strategy

for problem (1)-(8) is presented on Fig. 1.

To understand the concept presented on Fig. 1 it would

be helpfull to investigate, how an iteration is executed. At

l-iteration, variables pl are specified by NLP solver. In this

situation, when vaules of pl are known, one can treat DAE

system as an initial value problem and integrate (2)-(4) for-

ward in time for periods l = 1,...,N T . For these purposes

solver ode15s was used, which can solve index-1 DAE. Dif-

ferential state profile, algebraic state profile and control func-

tion profile were obtained as results of this step. Next com-

ponent evaluates the gradient of the objective and constraintfunctions with respect to pl. Because function and gradient

informations are passed to the NLP solver, then the decision

variables can be updated. The convergence of problem (9)-

(11) is depending on NLP solver. In [8] it is suggested, that

SQP codes, as NPSOL, NLPQP and fmincon are gen-

erally well suited for this task, as they require relatively few

iterations to converge. Because SQP-codes can suffer, when

an initial point is far from a solution [15], one decided to de-

sign the Two-Stage Solution Technique.

Fig. 1. Sequential dynamic optimization strategy.

Two-Stage Solution Technique and Multiple ShootingMethod

The idea of using the Multi-Stage Solution Technique

comes from premise, that a hard problem can be broken into

a sequence of easier subproblems. The second motivation

are results presented in the book [6]. The mentioned ap-

proach was used successfully for such problems as Optimal

Lunar Swingby Trajectory and In-Flight Dynamic Optimiza-

tion of Wing Trailing Egde Surface Positions. The first case

consisted of four steps: (1) Three-Impulse, Conic Solution,

(2) Three-Body Approximation to Conic Solution, (3) Optimal

Three-Body Solution with Fixed Swingby Time and (4) Opti-

mal Three-Body Solution. The second problem was solved

in 3 steps: (1) Reference Trajectory Estimation, (2) Aerody-

namic Drag Model Approximation and (3) Optimal Camber

Prediction.

The main idea is to use results from n − 1-step as an

initial solution in n-step.

For optimal control of Kinetic Batch Reactor it was de-

cided to design the Two-Stage Solution Technique. In thefirst stage an easier optimal control problem was investi-

gated. The problem was solved with ¨classical¨single shoot-

ing method. The continuity conditions for state variables and

control function are imposed explicite. Results obtained in

this stage are going to be the initial conditions for the sec-

ond stage. Because there are relatively few parameters in

this step and a computation effort is acceptable , one can

use active − set approach [14]. As it was investigated,

active − set approach gave good results for wide range of

problems and all iterations are feasible [16]. This step is

aimed to obtain continuous state trajectory and continuouscontrol tajectory as the solution. If it is possible, the most im-

portant constraints should be satisfied. The knowledge about

the systems is necessery to constitute a model and then to

design optimum batch process operating profiles. If ¨single

shooting method/" does not give any results, there has to be

used another method, ex. dichotomy approach [8].

Because in the second stage multiple shooting method

was used, the NLP solver has to obtain a solution of more

complex problem. One can hope, that initial conditions are

close enough to the optimal solution. The optimization al-

gorithm starts with initial conditions, which provide a feasible

initial solution. The new cost function have to reflect this so-

lution, so a new solution, which is not feasible can not be

accepted. This step is aimed at the determination a solu-tion satysiyng all the constraints. In other words, in this step

the optimization algorithm starts with continuous state and

control trajectories. Some constraints are satisfied as well.

Now the algorithm searches for a new solution. This is en-

abled by adding optimization variables and new cost function.

The first condition, which has to be satifiesd, is continuity of

state trajectory. This approach is in opposition to solution

presented in [6], where one of the obtained state trajectories

has discontinuities. We allowed discontinuities only in con-

trol function, but the differences betwen control functions in

grid points have to be minimized. Now it is expected, that ob-

tained solution have high practical meaning. As NLP solver

fmincon in MATLAB R2010b with the option Algorithmsqp [14] was proposed.

When the multiple shooting approach is used, the time

domain is partitioned into smaller time periods and the DAE

models are integrated separately in each element. To provide

continuity of the states across elements, equality constraints

are added to the nonlinear program. Inequality constraints

for states and controls are then imposed directly at the grid

points tl. Path constraints on state trajectory may be violated

between grid points [6, 8]. For optimal control problems with

the multiple shooting approach, the following formulation was

considered.

(12) φ( p) → min,

subject to

(13) yl−1(tl−1)− yl0 = 0, l = 2, ...N T ,

(14) yN T (tN T )− yf = 0, yl(0) = yl0,

(15) plL ≤ pl ≤ plU

(16) zl

L ≤ zl

(tl) ≤ zl

U

(17) ylL ≤ yl(tl) ≤ ylU , l = 1,...,N T ,

PRZEGL AD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 88 NR 6 / 2012 177



Table 1. Batch reactor dynamic variables.

Description

y0 Differential state [HA] + [A−]y1 Differential state [BM ]y2 Differential state [HABM ] + [ABM −]y3 Differential state [AB]y4 Differential state [MBMH ] + [MBM −]y5 Differential state [M −]z1 Algebraic state −log([H +])z2 Algebraic state [A−]z3 Algebraic state [ABM −]z4 Algebraic state [MBM −]z5 Reaction Temperature

with the DAE system

(18)dyi

dt(t) = f l(yl(t), zl(t), pl), yl(tl−1) = yl0,

(19) gl(yl(t), zl(t), pl), t ∈ (tl−1, tl], l = 1,...,N T .

Description of Kinetic Batch Reactor model

The Two-Stage Solution Technique was tested with the

model of Kinetic Batch Reactor. There is a detailed descrip-

tion of the Kinetic Batch Reactor model in the book [6]. Here

there are presented only these parts, which enables to write

and to use the software and then to solve optimal control

problem.

In the reactor the desired product AB is formed in the

reaction

(20) HA + 2BM → AM + MBMH.

Here a kinteic model of the batch reactor system is presented

in the same manner like in the book [6]. There are distin-guished differential and algebraic states. Differential and al-

gebraic variables are denoted as yj and zj , respectively. All

of the species concentrations are given in mol per kg of the

reaction mixture. There are batch reactor dynamic varibles in

the table 1 and six differential mass balance equations in the

model

(21) y1 = −k2y2z2,

(22) y2 = −k1y2y6 + k−1z4 − k2y2z2,

(23) y3 = k2y2z2 + k3y4y6 − k−3z3,

(24) y4 = −k3y4y6 + k−3z3,

(25) y5 = k1y2y6 − k−1z4,

(26) y6 = −k1y2y6 + k−1z4 − k3y4y6 + k−3z3.

The electroneutrality condition

(27) 0 = p− y6 + 10−z1 − z2 − z3 − z4,

There are three equilibrium condition

(28) 0 = z2 −K 2y1

K 2 + 10−

z1

,

(29) 0 = z3 −K 3y3

K 3 + 10−z1,

Table 2. Values of parameters

k1 = 1.3708× 1012, k−1 = 1.6215× 1020,

k2 = 5.2282× 1012,

β 1 = 9.2984× 103, β −1 = 1.3108× 104,β 2 = 9.5999× 103,

K 1 = 2.575× 10−16, K 2 = 4.876× 10−14,K 3 = 1.7884× 10−16.

(30) 0 = z4 −K 1y5

K 1 + 10−z1,

where

(31) k1 = k1exp(−β 1/z5),

(32) k−1 = k−1exp(−β −1/z5),

(33) k2 = k2exp(−β 2/z5),

(34) k3 = k1,

(35) k−3 =1

2k−1.

There are the values for the parameters kj (kg / gmol / hr ),

β j ( K ), and K j ( gmol / kg) in the table 2. As the control

variable we treat the reaction temperature T (K ). For the

duration of the process the control variable is limited to

(36) 293.15 ≤ z5(t) ≤ 393.15.

One of the designed parameters is the initial catalyst concen-

tration p(gmol / kg). This parameter is treated as an optimiza-tion variable and is restricted by the bounds

(37) 0 ≤ p ≤ 0.0262.

At the initail time t = 0 the initial catalyst concentration is re-

alted to the corresponding state through the point constraint

(38) = y6(0)− p = 0.

From [6] it is known, that the remaining states are fixed by the

boundary conditions

(39) y1(0) = 1.5776,

(40) y2(0) = 8.32,

(41) y3(0) = y4(0) = y5(0) = 0.

It is required, that at the free final time

(42) y4(tF ) ≥ 1.

There is the nonlinear inequality path constraint in order to

restrict the rate of product formation during the initial 25% of

the process time. For 0 ≤ t ≤ (tF /4)

(43) y4(t) ≥ at2,

where a = 2(gmol / kg / hr · hr).

The desired objective function is to minimize the quantity

(44) F = γ 1tF + γ 2 p,

where γ 1 = 1 and γ 2 = 100.




ResultsTo solve problem stated above, the Two-Stage Solution

Technique was used. All calculations were performed on the

processor Intel(R) Core(TM) i5 CPU 2.67 GHz.

In the first stage the single shooting method and lin-

ear control function were used. State variables and control

function are continuous. We have to minimize augmented

cost function, which were obtained as result of scalar prod-uct method with penalty function approach. Augmented cost

function consists of added primary cost function F (44), final

condition (42) and nonlinear inequality path constraint(43).

To make the new cost function smooth, for last two terms the

smoothing patrameter were used.

Three control functions were used.

(45) u5,1 = a1t + b1

for the first stage,

(46) u5,2 = a2t + b2

for the second stage and

(47) u5,3 = a3t + b3

for the last stage. Because control function is continuous,

parameters b2 and b3 can be obtained analytically. Vector p1was constituted as below

(48) p1 = [a1, b1, a2, a3, p , tf ]T .

Initial conditions

(49) p1(0) = [1000, 293.15, 100, 100, 0.0262, 3.0]T ,

lower and upper bounds

(50) lb1 = [−1000, 293.15,−100,−100, 0, 1.0]T ,

(51) ub1 = [1000, 393.15, 100, 100, 0.0262, 3.0]T .

The results, which were obtained after 537.3142 CPU

time, are presented on the Fig. 2 - Fig. 7. There is the control

function trajectory on the Fig. 8. State and control trajectories

were denoted with the dotted line.

¨Active − set" with the options options1 was used as

an optimization algorithm.

options1 = optimset(TolFun, 1e− 7, TolX , 1e−7, Algorithm, active − set, Hessian, f in − diff −

grads

,

MaxFunEvals

, 200000,

Display

,

iter−

detailed);In the second step the option sqp as Algorithm was

used. Results from previous step were used as initial con-

diotions. The obtained NLP problem is more complex, but

now with good initial conditions, fast convergence can be ex-

pected.

Three control functions were used.

(52) U 5,1 = A1t + B1

for the first stage,

(53) U 5,2 = A2t + B2

for the second stage and

(54) U 5,3 = A3t + B3

for the last stage.

Because mulitple shooting method was used, control

function is not continuous and parameters b2 and b3 will not

be obtained analytically. Vector p2 was constituted as below

(55) p2 = [B1, B2, B3, p , tf , A1, A2, A3]T .

Initial conditions

(56) p2(0) = [b1, 393.15, 393.15, p , tf , a1, a2, a3]T ,

lower and upper bounds

(57)

lb2 = [293.15, 293.15, 293.15, 0, 1.0,−1000,−100,−100]T ,

(58)

up2 = [393.15, 393.15, 393.15, 0.0262, 3.0, 1000, 100, 100]T .

The results were obtained after 7613.7 CPU time. They

were denoted on the Fig. 2 - Fig. 7 and Fig. 8 with the solid

line.

The optimization algorithm ¨sqp" with the options

options2 was usedoptions2 = optimset(TolFun, 1e− 7, TolX , 1e−

7, Algorithm, sqp, Hessian, f in − diff −

grads, MaxFunEvals, 200000, Display, iter −

detailed);In both stages solver ode15s used Backward Differential

Formula.

Fig. 2. Differential state Y1.


ConclusionIn the article a new approach for optimal control of a

class of DAE systems were presented. The Two-Stage So-lution Technique enables to use one of the general-purpose

NLP solvers. Because the Matlab is commonly known among

engineers, one can expect, that this approach will be useful

PRZEGL AD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 88 NR 6 / 2012 179





for finding solutions of the optimal control problems of real-

life systems. The efficiency of this method was tested on the

Kinetic Batch Reactor model as an example of the stiff and

highly nonlinear DAE system. This model emphasizes the

use of equilibrium constraints of various physical types, bothelectrical and chemical.

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Authors: M.Sc. Paweł Drag, Prof. Krystyn Styczen,

Institute Computer Engineering, Control and Robotics,

Wrocław University of Technology, Janiszewskiego 17, 50-

372 Wrocław, Poland, email: [email protected],

[email protected].


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A Two-Step Approach for Optimal Control of Kinetic Batch